Let's analyze each game to determine whether the empirical probability (based on Jason's recorded data) aligns closely with the theoretical probability of winning that game.
1. Game A: Carnival Wheel
- Theoretical probability of winning (landing on the violet section): [tex]\( \frac{1}{7} \)[/tex].
- Number of wins: 26, Number of losses: 149.
- Total number of attempts: [tex]\( 26 + 149 = 175 \)[/tex].
- Empirical probability of winning: [tex]\( \frac{26}{175} \approx 0.1486 \)[/tex].
Given data indicates that this empirical probability does indeed align closely with the theoretical probability.
Therefore, the statement for Game A is:
``true``
2. Game B: Bingo Ball
- Theoretical probability of winning (picking the white ball): [tex]\( \frac{1}{10} \)[/tex].
- Number of wins: 82, Number of losses: 118.
- Total number of attempts: [tex]\( 82 + 118 = 200 \)[/tex].
- Empirical probability of winning: [tex]\( \frac{82}{200} = 0.41 \)[/tex].
Given data indicates that this empirical probability does not align closely with the theoretical probability.
Therefore, the statement for Game B is:
``false``
3. Game C: Dice to Win
- Theoretical probability of winning (rolling a 6): [tex]\( \frac{1}{6} \)[/tex].
- Number of wins: 39, Number of losses: 201.
- Total number of attempts: [tex]\( 39 + 201 = 240 \)[/tex].
- Empirical probability of winning: [tex]\( \frac{39}{240} \approx 0.1625 \)[/tex].
Given data indicates that this empirical probability does indeed align closely with the theoretical probability.
Therefore, the statement for Game C is:
``true``
Thus, the final answers are:
- The results from Game [tex]\(A\)[/tex] align closely with the theoretical probability of winning that game. true
- The results from Game [tex]\(B\)[/tex] align closely with the theoretical probability of winning that game. false
- The results from Game [tex]\(C\)[/tex] align closely with the theoretical probability of winning that game. true
